Problem 1. (computer) (0 points - do not return)

Write a program which uses the bisection method for locating roots of equations. Use the

program to find the root of the following equation

9x

4 +18x

3 +38x

2 −57x+14 = 0

in the interval [0,1].

Problem 2. (computer) (3 points)

(a) Write a program which uses the Newton’s method for locating roots of equations. Use

the program to find the root of the following equation

x

3 −x−5 = 0

Use the initial point x0 = 0.57735. Limit your iterations to 50. Print out the results and

explain them.

(b) Construct a hybrid method which utilizes a combination of the bisection and Newton’s

algorithms to ensure global convergence. This algorithm takes a bisection step whenever

Newton’s algorithm would take the solution out of bounds. As input, this method needs

two numbers a1 and a2 which bracket the root and the starting point for the Newton’s

method x0 (this can, of course, be computed from the brackets, but for testing purposes

use the value x0 = 0.57735).

Hint: Modify your code from part (a) by inserting a condition to use bisection if Newton

is out of range.

1

Problem 3. (computer / Matlab) (3 points)

Basin of attraction. Consider the complex polynomial z

3 −1. Its roots are the three cube

roots of unity. Generate a picture showing the three basins of attraction in the square

region defined by −1 ≤ Real(z) ≤ 1 and −1 ≤ Imag(z) ≤ 1.

To do this, use a mesh of 1000×1000 pixels inside the square. The center point of each

pixel is used to start the iteration of Newton’s method. Assign a particular basin color to

each pixel if convergence to a root is obtained with a maximum of 100 iterations.

In order to limit the large number of iterations use a criterion for stopping the iteration

when it gets within a certain neighborhood of the root. The criterion for convergence is

to check both |zn+1 −zn| < ε and |z

3

n+1 −1| < ε with a small value such as ε = 10−12 as

well as using a maximum number of steps.

Hint: It is best to test your program and to get a crude picture with only a small number

of pixels such as 10×10.

Note: For this problem, it is permitted to use Matlab (solutions in other programming

languages are of course also accepted).

2